Given a monotone graph property P , consider p (P ), the probability that a random graph with edge probability p will have P . The function d p (P )=dp is the key to understanding the threshold behavior of the property P . We show that if d p (P )=dp is small (corresponding to a non-sharp threshold), then there is a list of graphs of bounded size such that P can be approximated by the property of having one of the graphs as a subgraph. One striking consequences of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of n.

Motivation: Our goal is to construct a model for genetic regulatory networks such that the model class: (i ) incorporates rule-based dependencies between genes; (ii ) allows the systematic study of global network dynamics; (iii ) is able to cope with uncertainty, both in the data and the model selection; and (iv ) permits the quantification of the relative influence and sensitivity of genes in their interactions with other genes.

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [25]. This implies that if the Unique Games

Abstract. In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let Vn(p) ={0,1} n denote the Hamming space endowed with the probability measure µp defined by µp(ɛ1,ɛ2,...,ɛn) = pk ·(1 − p) n−k,where k = ɛ1+ ɛ2+ ···+ ɛn. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on {1, 2,...,n} such that A is invariant under Γ. Theorem. For every symmetric monotone A,ifµp(A)>ɛthen µq(A)> 1−ɛ for q = p + c1 log(1/2ɛ) / log n. (c1isan absolute constant.) 1. Graph properties A graph property is a property of graphs which depends only on their isomorphism class. Let P be a monotone graph property; that is, if a graph G satisfies P

Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 De-randomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 De-randomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers. 1 Introduction Computational Complexity is the subfield of Theoretical Computer Science that aims to understand "how much" computation is necessary and sufficient to perform certain computational tasks. For example, given a computational problem it tries to establish tight upper and lower bounds on the length of the computation (or on other resources, like space). Unfortunately, for many, practically relevant, computational problems no tight bounds are known. An illustrative example is the well known P versus NP problem: for all NP-complete problems the current upper and lower bounds lie exponentially ...

We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for picking highly correlated samples from a finite set, with the property that the probability of hitting any sufficiently large subset is high. These procedures require a relatively small number of random bits and are robust with respect to the quality of the random bits. Using these sampling procedures to sample random inputs of polynomial time probabilistic algorithms, we can simulate the performance of some probabilistic algorithms with less random bits or with low quality random bits. We obtain the following results: 1. The error probability of an RP or BPP algorithm that operates with a constant error bound and requires n random bits, can be made exponentially small (i.e. 2 \Gamman ), with only (3 + ffl)n random bits, as opposed to standard amplification techniques that require \Omega\Gamma n 2 ) random bits for the same task. This result is nearly optimal, since the informati...

Recently there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Rather intruigingly, experimental results with various models for generating random CSP instances suggest a "threshold-like" behaviour and some theoretical work has been done in analyzing these models when the number of variables is asymptotic. In this paper we show that the models commonly used for generating random CSP instances suffer from a wrong parameterization which makes them unsuitable for asymptotic analysis. In particular, when the number of variables becomes large almost all instances they generate are, trivially, overconstrained. We then present a new model that is suitable for asymptotic analysis and, in the spirit of random SAT, we derive lower and upper bounds for its parameters so that the instances generated are "almost surely" over- and underconstrained, respectively. Finally, we apply the technique introduced in [19], to one of the popular models in Artificial Intelligence and derive sharper estimates for the probability of being overconstrained as a function of the number of variables. 1

The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NP-hard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1

For any fixed 6> 0, we show how to simulate RP algorithms in time nO(‘Ogn) using the output of a 6-source wath min-entropy R‘. Such a weak random source is asked once for R bits; it outputs an R-bit string such that any string has probability at most 2-Rc. If 6> 1- l/(k + l), our BPP simulations take time no(‘og(k)n) (log(k) is the logarithm iterated k times). We also gave a polynomial-time BPP simulation using Chor-Goldreich sources of min-entropy Ro(’), which is optimal. We present applications to time-space tradeoffs, expander constructions, and the hardness of approximation. Also of interest is our randomness-efficient Leflover Hash Lemma, found independently by Goldreich & Wigderson.